Timeline for Minimal resolution of singularities of surfaces
Current License: CC BY-SA 4.0
3 events
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| May 3, 2023 at 20:01 | comment | added | R. van Dobben de Bruyn | If $\pi^{-1}(x) = \{y\}$, then $Y \times_X \operatorname{Spec} \mathcal O_{X,x} \cong \operatorname{Spec} \mathcal O_{Y,y}$, so $\mathcal O_{X,x} \to \mathcal O_{Y,y}$ is finite (since it is proper and quasi-finite). Since it is also birational extension, normality of $\mathcal O_{X,x}$ shows that $\mathcal O_{X,x} \stackrel\sim\to \mathcal O_{Y,y}$. | |
| May 3, 2023 at 12:39 | comment | added | Jérémy Blanc | Thanks for your answer. Why is $\pi^{-1}(x)$ never a point for $x\in X$ singular? [Hartshorne, Cor. III.11.5] shows that the preimage is connected, but why not only one point? Can we show that we would have a local isomorphism in this case? | |
| May 2, 2023 at 17:43 | history | answered | R. van Dobben de Bruyn | CC BY-SA 4.0 |